The document summarizes Christophe Paul's work on algorithms for modular decomposition. It discusses Ehrenfeucht et al's modular decomposition algorithm which computes the modular partition M(G,v) of a graph G with respect to a vertex v. It then computes the modular decomposition of the quotient graph G/M(G,v) and the induced subgraphs G[X] for each module X in M(G,v). The document also discusses Gallai's theorem on the decomposition of graphs into parallel, series, or prime modules and algorithms for recognizing cographs and computing the modular decomposition tree.

1. Algorithmics of Modular
decomposition
Christophe Paul
CNRS - LIRMM, Montpellier France
Algorithms & Permutations Workshop
February 20, 2012
Joint work with: A. Bergeron, S. B´rard, S. Bessy, B.M. Bui Xuan, C.
e
Chauve, D. Corneil, F. Fomin, E. Gioan, M. Habib, A. Perez, S. Saurabh,
S. Thomass´, M. Tedder, L. Viennot. . .
e

2. Modular decomposition of undirected graphs
Ehrenfeucht et al’s modular decomposition algorithm
Common Intervals of permutations
Modular decomposition of tournaments
Kernelization algorithm for FAST

3. Permutations and permutation graphs
1 2 3 4 5 6 7 8 9 10 11
7 6 1 5 10 8 11 9 2 3 4
5
2 8
6 10
1
3
7 9
4 11

4. Permutations and permutation graphs
1 2 3 4 5 6 7 8 9 10 11
7 6 1 5 10 8 11 9 2 3 4
5
2 8
6 10
1
3
7 9
4 11

5. Permutations and permutation graphs
1 2 3 4 5 6 7 8 9 10 11
7 6 1 5 10 8 11 9 2 3 4
5
2 8
6 10
1
3
7 9
4 11
Does a permutation graph have a unique representation ?

6. Permutations and permutation graphs
1 2 3 4 5 6 7 10 8 11 9
7 6 1 5 8 9 10 11 2 3 4
5
2 8
6 10
1
3
7 9
4 11
Does a permutation graph have a unique representation ?

7. Modules
A subset of vertices M of a graph G = (V , E ) is a module iff
∀x ∈ V M, either M ⊆ N(x) or M ∩ N(x) = ∅
5
2 8
6 10
1
3
7 9
4 11

8. Modules
A subset of vertices M of a graph G = (V , E ) is a module iff
∀x ∈ V M, either M ⊆ N(x) or M ∩ N(x) = ∅
5
2 8
6 10
1
3
7 9
4 11
Examples of modules:
connected components
connected components of G

9. Modules
A subset of vertices M of a graph G = (V , E ) is a module iff
∀x ∈ V M, either M ⊆ N(x) or M ∩ N(x) = ∅
5
2 8
6 10
1
3
7 9
4 11
A graph (a module) is prime is all its modules are trivial:
e.g. the P4 .

10. Modules
A subset of vertices M of a graph G = (V , E ) is a module iff
∀x ∈ V M, either M ⊆ N(x) or M ∩ N(x) = ∅
5
2 8
6 10
1
3
7 9
4 11
A graph (a module) is prime is all its modules are trivial:
e.g. the P4 .
A graph (a module) is degenerate if every subset of vertices is
a module: cliques and stables.

11. Permutation graph recognition
Theorem [Gallai’67]
A permutation graph has a unique representation (up to reversal)
iff it is prime.

12. Permutation graph recognition
Theorem [Gallai’67]
A permutation graph has a unique representation (up to reversal)
iff it is prime.
Recognition algorithm
Recursively solve the problem on modules
Solve the prime case (with linear time transitive orientation
algorithm)

13. Permutation graph recognition
Theorem [Gallai’67]
A permutation graph has a unique representation (up to reversal)
iff it is prime.
Recognition algorithm
Recursively solve the problem on modules
Solve the prime case (with linear time transitive orientation
algorithm)
Theorem [McConnell and Spinrad’99]
The permutation graph recognition problem can be solved in
O(n + m) time
we need a linear time modular decomposition algorithm

14. Partitive families
If M and M are two overlapping modules then

15. Partitive families
If M and M are two overlapping modules then
M M is a module

16. Partitive families
If M and M are two overlapping modules then
M M is a module
M ∩ M is a module

17. Partitive families
If M and M are two overlapping modules then
M M is a module
M ∩ M is a module
M∆M is a module
The set of modules of a graph forms a partitive family
A module is strong if it does not overlap any other module

18. Partitive families
If M and M are two overlapping modules then
M M is a module
M ∩ M is a module
M∆M is a module
The set of modules of a graph forms a partitive family
A module is strong if it does not overlap any other module
5
2 8
1 2 3 4 5 6 7 8 9 10 11
6 10
1
234 67 8 9 10 11
3
7 9 1 2 3 4 5 6 7 8 9 10 11
4 11
Strong modules are nested into an inclusion tree: the modular
decomposition tree MD(G )

19. Modular partition and quotient graph
A partition P of the vertex set of a graph G is a modular partition
if every part is a module of G .
5
2 8
6 10
1
3
7 9
4 11

20. Modular partition and quotient graph
A partition P of the vertex set of a graph G is a modular partition
if every part is a module of G .
5
2 8
6 10
1
3
7 9
4 11
If P is a modular partition of G , the quotient graph G/P is the
induced subgraph obtained by choosing one vertex per part of P.

21. Theorem [Gal’67,CHM81]
Let G = (V , E ) be a graph. Then either
1. (parallel) G is not connected, or
2. (series) G is not connected, or

22. Theorem [Gal’67,CHM81]
Let G = (V , E ) be a graph. Then either
1. (parallel) G is not connected, or
2. (series) G is not connected, or
3. (prime) G/M(G ) is a prime graph, with M(G ) the modular
partition containing the maximal strong modules of G .
5 Prime
2 8 1 2 3 4 5 6 7 8 9 10 11
6 10
1
parallel series Prime
3 234 67 8 9 10 11
7 9
4 11 1 2 3 4 5 6 7 8 9 10 11

23. Theorem [Gal’67,CHM81]
Let G = (V , E ) be a graph. Then either
1. (parallel) G is not connected, or
2. (series) G is not connected, or
3. (prime) G/M(G ) is a prime graph, with M(G ) the modular
partition containing the maximal strong modules of G .
5 Prime
2 8 1 2 3 4 5 6 7 8 9 10 11
6 10
1
parallel series Prime
3 234 67 8 9 10 11
7 9
4 11 1 2 3 4 5 6 7 8 9 10 11
Observation: If a P4 on {a, b, c, d} overlap a module M, then
|M ∩ {a, b, c, d}| = 1

24. Modular decomposition algorithms
O(n4 ) [Cowan, James, Stanton’72]
O(n3 ) [Blass, 1978], [Habib, Maurer’79]
O(n2 ) [McConnell, Spinrad’89]

25. Modular decomposition algorithms
O(n4 ) [Cowan, James, Stanton’72]
O(n3 ) [Blass, 1978], [Habib, Maurer’79]
O(n2 ) [McConnell, Spinrad’89]
O(n + mα(m, n)) [Spinrad’92], [Cournier, Habib’93]
O(n + m) [McConnell, Spinrad’94], [Cournier, Habib’94]

26. Modular decomposition algorithms
O(n4 ) [Cowan, James, Stanton’72]
O(n3 ) [Blass, 1978], [Habib, Maurer’79]
O(n2 ) [McConnell, Spinrad’89]
O(n + mα(m, n)) [Spinrad’92], [Cournier, Habib’93]
O(n + m) [McConnell, Spinrad’94], [Cournier, Habib’94]
O(n + m log n) [Habib, Paul, Viennot’99] (factoring
permutation), [McConnell, Spinrad’00]

27. Modular decomposition algorithms
O(n4 ) [Cowan, James, Stanton’72]
O(n3 ) [Blass, 1978], [Habib, Maurer’79]
O(n2 ) [McConnell, Spinrad’89]
O(n + mα(m, n)) [Spinrad’92], [Cournier, Habib’93]
O(n + m) [McConnell, Spinrad’94], [Cournier, Habib’94]
O(n + m log n) [Habib, Paul, Viennot’99] (factoring
permutation), [McConnell, Spinrad’00]
O(n + m) [Capelle, Habib’97] (factoring permutation)
[Dahlhaus, Gustedt, McConnell’97], [Tedder, Corneil, Habib,
Paul’08]
other many others for variants of modular decomposition

28. Cographs - Totally decomposable graphs
Theorem: A graph is a cograph (a P4 -free graph 1 2 3 4)
iff its modular decomposition tree does not contain any prime node
b
u Series
d
v Parallel Parallel
c
e Series Series e Series Series
a
w Parallel z d Parallel y w u v
z
y x a c b
x

29. Cographs - Totally decomposable graphs
Theorem: A graph is a cograph (a P4 -free graph 1 2 3 4)
iff its modular decomposition tree does not contain any prime node
b
u Series
d
v Parallel Parallel
c
e Series Series e Series Series
a
w Parallel z d Parallel y w u v
z
y x a c b
x
Cographs can be built from the single vertex with the disjoint
union and series composition
Exercice: prove that cographs are permutation graphs

30. Cographs - Totally decomposable graphs
Theorem: A graph is a cograph (a P4 -free graph 1 2 3 4)
iff its modular decomposition tree does not contain any prime node
b
u Series
d
v Parallel Parallel
c
e Series Series e Series Series
a
w Parallel z d Parallel y w u v
z
y x a c b
x
Linear time recognition algorithms
incremental [Corneil, Pearl, Stewart’85]
partition refinement [Habib, P.’05]
LexBFS [Bretscher, Corneil, Habib, P.’08]

31. Modular decomposition of undirected graphs
Ehrenfeucht et al’s modular decomposition algorithm
Common Intervals of permutations
Modular decomposition of tournaments
Kernelization algorithm for FAST

32. Ehrenfeucht et al’s modular decomposition algorithm
M(G , v ) is the modular partition composed by
{v } and the maximal modules of G not containing v .
series
prime parallel 8
9
8
9 A
@ A
@ C
B C
B
¡
¡
parallel series
series ! ! #
#
4
5
4
5 7
6 7
6
2
3
2
3
0
1
0
1
series ¢
£
¢
£
¨
©
¨
©
¦
§
¦
§
$
%
$
%
(
)
(
)
v ¤
¥
¤
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v
'
'
1. Compute M(G , v )
2. Compute MD(G/M(G ,v ) )
3. For each X ∈ M(G , v ) compute MD(G [X ])

33. Computation of M(G , v ) (1)
Lemma [MR84] Let P be a modular partition of G = (V , E ).
X ⊆ P is a module of G/P iff ∪M∈X M is a module of G .
5
2 8
6 10
1
3
7 9
4 11

34. Computation of M(G , v ) (1)
Lemma [MR84] Let P be a modular partition of G = (V , E ).
X ⊆ P is a module of G/P iff ∪M∈X M is a module of G .
A vertex x is a splitter for a set S of vertices if
∃y , z ∈ S with xy ∈ S and xz ∈ E /
We say that x separate y and z.
2
6
1 4
3
7

35. Computation of M(G , v ) (1)
Lemma [MR84] Let P be a modular partition of G = (V , E ).
X ⊆ P is a module of G/P iff ∪M∈X M is a module of G .
A vertex x is a splitter for a set S of vertices if
∃y , z ∈ S with xy ∈ S and xz ∈ E /
We say that x separate y and z.
2
6 Lemma If x is a splitter for the set
1 4 S, then any module M containing S
must also contain x.
3
7

36. Computation of M(G , v ) (2)
Lemma If v is a splitter of a set S, then for any module M ⊆ S
either M ⊆ S ∩ N(v ) or M ⊆ M ∩ N(v )

37. Computation of M(G , v ) (2)
Lemma If v is a splitter of a set S, then for any module M ⊆ S
either M ⊆ S ∩ N(v ) or M ⊆ M ∩ N(v )
O(n + m log n) time using vertex partitioning algorihtm
N(v) N(v)
v

38. Computation of M(G , v ) (2)
Lemma If v is a splitter of a set S, then for any module M ⊆ S
either M ⊆ S ∩ N(v ) or M ⊆ M ∩ N(v )
O(n + m log n) time using vertex partitioning algorihtm
v x

39. Computation of M(G , v ) (2)
Lemma If v is a splitter of a set S, then for any module M ⊆ S
either M ⊆ S ∩ N(v ) or M ⊆ M ∩ N(v )
O(n + m log n) time using vertex partitioning algorihtm
v x y

40. Computation of M(G , v ) (2)
Lemma If v is a splitter of a set S, then for any module M ⊆ S
either M ⊆ S ∩ N(v ) or M ⊆ M ∩ N(v )
O(n + m log n) time using vertex partitioning algorihtm
v z

41. Computation of MD(G/M(G ,v ) ) (3)
The modules of G/M(G ,v ) are linearly nested:
any non-trivial module contains v
6
3 4 5
2
v 1

42. Computation of MD(G/M(G ,v ) ) (3)
The modules of G/M(G ,v ) are linearly nested:
any non-trivial module contains v
The forcing graph F(G , v ) has edge − iff y separates x and v
→
xy
6
5
4
6
3
3 4 5
6 v
2
2
1
v 1
5 4 3 2 1

43. Computation of MD(G/M(G ,v ) ) (3)
The modules of G/M(G ,v ) are linearly nested:
any non-trivial module contains v
The forcing graph F(G , v ) has edge − iff y separates x and v
→
xy
6
5
4
6
3
3 4 5
6 v
2
2
1
v 1
5 4 3 2 1

44. Computation of MD(G/M(G ,v ) ) (4)
Complexity
[Ehrenfeucht et al.’94] gives a O(n2 ) complexity.
[MS00]: simple O(n + m log n) vertex partitioning algorithm
[DGM’01]: O(n + m.α(n, m)) and a more complicated
O(n + m) implementation.
Other algorithms
[CH94] and [MS94]: the first linear algorithms.
[MS99]: O(n + m) algorithm which extends to transitive
orientation.

45. Computation of MD(G/M(G ,v ) ) (4)
Complexity
[Ehrenfeucht et al.’94] gives a O(n2 ) complexity.
[MS00]: simple O(n + m log n) vertex partitioning algorithm
[DGM’01]: O(n + m.α(n, m)) and a more complicated
O(n + m) implementation.
Other algorithms
[CH94] and [MS94]: the first linear algorithms.
[MS99]: O(n + m) algorithm which extends to transitive
orientation.
[Spinrad’03] The new [linear time] algorithm [MS99] is currently
too complex to describe easily [...] I hope and believe that in a
number of years the linear algorithm can be simplified as well.
[Tedder, Corneil, Habib, P.’08] simple linear time algorithm

46. Modular decomposition of undirected graphs
Ehrenfeucht et al’s modular decomposition algorithm
Common Intervals of permutations
Modular decomposition of tournaments
Kernelization algorithm for FAST

47. Back to permutations: common intervals
In a permutation σ, a set S of consecutive elements is called an
interval. Likewise a set S is a common interval of several
permutations σ1 , σ2 . . . if it is an interval for every σi .
1 2 3 4 5 6 7 8 9 10 11
7 6 1 5 10 8 11 9 2 3 4

48. Back to permutations: common intervals
In a permutation σ, a set S of consecutive elements is called an
interval. Likewise a set S is a common interval of several
permutations σ1 , σ2 . . . if it is an interval for every σi .
5
2 8
1 2 3 4 5 6 7 8 9 10 11
6 10
1
3
7 6 1 5 10 8 11 9 2 3 4 7 9
4 11
Obs.: common intervals of σ1 , σ2 are not modules of G (σ1 , σ2 )

49. Back to permutations: common intervals
In a permutation σ, a set S of consecutive elements is called an
interval. Likewise a set S is a common interval of several
permutations σ1 , σ2 . . . if it is an interval for every σi .
5
2 8
1 2 3 4 5 6 7 8 9 10 11
6 10
1
3
7 6 1 5 10 8 11 9 2 3 4 7 9
4 11
Obs.: common intervals of σ1 , σ2 are not modules of G (σ1 , σ2 )
Computation of all common intervals in linear time - O(n2 ) -
[Uno, Yagura’00]

50. Back to permutations: common intervals
In a permutation σ, a set S of consecutive elements is called an
interval. Likewise a set S is a common interval of several
permutations σ1 , σ2 . . . if it is an interval for every σi .
5
2 8
1 2 3 4 5 6 7 8 9 10 11
6 10
1
3
7 6 1 5 10 8 11 9 2 3 4 7 9
4 11
Obs.: common intervals of σ1 , σ2 are not modules of G (σ1 , σ2 )
Strong (common) interval doesn’t overlap other common intervals
Lemma [de Montgolfier] A set S is a strong interval of σ1 and σ2
iff it is a strong module of the permutation graph G (σ1 , σ2 )

51. Common intervals (2)
The family of common intervals is weakly partitive:
7 1 6 5 4 3 2 9 8 11 10
if I1 and I2 are two common intervals then
I1 ∪ I2 is a common interval
I1 ∩ I2 is a common interval
I1 I2 and I2 I1 are common intervals

52. Common intervals (2)
The family of common intervals is weakly partitive:
7 1 6 5 4 3 2 9 8 11 10
7 1 5 3 8
if I1 and I2 are two common intervals then
I1 ∪ I2 is a common interval
I1 ∩ I2 is a common interval
I1 I2 and I2 I1 are common intervals
Let I be a partition of [1 . . . n] into common intervals of the
permutation σ, then we denote by σ/I the quotient permutation
defined on [1 . . . |I|]

53. Common intervals (2)
The family of common intervals is weakly partitive:
7 1 6 5 4 3 2 9 8 11 10
4 1 3 2 5
if I1 and I2 are two common intervals then
I1 ∪ I2 is a common interval
I1 ∩ I2 is a common interval
I1 I2 and I2 I1 are common intervals
Let I be a partition of [1 . . . n] into common intervals of the
permutation σ, then we denote by σ/I the quotient permutation
defined on [1 . . . |I|]

54. Common intervals (3)
Theorem Let σ be a permutation on [1, . . . n] and I be the
partition into maximal common intervals of σ, then either
1. σ/I = 1|I| - the identity on [1 . . . |I|]
2. σ/I = 1|I| - the reverse identity on [1 . . . |I|]
3. σ/I is prime - or simple (does not have non-trivial common
interval)

55. Common intervals (3)
Theorem Let σ be a permutation on [1, . . . n] and I be the
partition into maximal common intervals of σ, then either
1. σ/I = 1|I| - the identity on [1 . . . |I|]
2. σ/I = 1|I| - the reverse identity on [1 . . . |I|]
3. σ/I is prime - or simple (does not have non-trivial common
interval)
7 1 6 5 4 3 2 9 8 11 10
Theorem (see e.g. [Bergeron et al.’08])
The common interval tree can be computed in O(n) time.

56. Common intervals (4)
A permutation σ is separable if it does not contains the pattern
3142
7 1 6 5 4 3 2 9 8 11 10

57. Common intervals (4)
A permutation σ is separable if it does not contains the pattern
3142
1 2 3 4
3 1 4 2 corresponds to the P4
7 1 6 5 4 3 2 9 8 11 10

58. Common intervals (4)
A permutation σ is separable if it does not contains the pattern
3142
1 2 3 4
3 1 4 2 corresponds to the P4
a permutation is separable iff
its common interval tree does not contains prime nodes
a permutation is separable iff
the permutation graph G (σ, 1) is a cograph (P4 -free graph)
7 1 6 5 4 3 2 9 8 11 10

59. Modular decomposition of undirected graphs
Ehrenfeucht et al’s modular decomposition algorithm
Common Intervals of permutations
Modular decomposition of tournaments
Kernelization algorithm for FAST

60. Modular decomposition of tournaments
1 2
8 3
7 4
6 5

61. Modular decomposition of tournaments
1 2
8 3
7 4
6 5
A module in a tournament is a set S such that for every x ∈ S
/
either ∀y ∈ S, x → y or ∀y ∈ S, y → x

62. Modular decomposition of tournaments
1 2
8 3
7 4
6 5
A tournament is transitive if there exists a permutation σ of V (T )
with no backward arcs
Theorem: Let T be a tournament and M(T ) be the modular
partition into maximal strong modules, then
1. either T/M(T ) is transitive - contains no backward arc
2. or T/M(T ) is prime

63. Modular decomposition of tournaments
1 2
8 3
7 4
6 5
1 2 3 4 5 6 7 8
A tournament is transitive if there exists a permutation σ of V (T )
with no backward arcs
Theorem: Let T be a tournament and M(T ) be the modular
partition into maximal strong modules, then
1. either T/M(T ) is transitive - contains no backward arc
2. or T/M(T ) is prime

64. Modular decomposition algorithm for tournaments
A factoring permutation of a tournament T (or a graph) is a
permutation σ of its vertices such that
every (strong) module of T is an interval of σ

65. Modular decomposition algorithm for tournaments
A factoring permutation of a tournament T (or a graph) is a
permutation σ of its vertices such that
every (strong) module of T is an interval of σ
Factoring permutation via a partition refinement algorithm in
linear time [de Mongolfier’03]
1 2 3 4 5 6 7 8

66. Modular decomposition algorithm for tournaments
A factoring permutation of a tournament T (or a graph) is a
permutation σ of its vertices such that
every (strong) module of T is an interval of σ
Factoring permutation via a partition refinement algorithm in
linear time [de Mongolfier’03]
3 4 5 6 1 2 7 8

67. Modular decomposition algorithm for tournaments
A factoring permutation of a tournament T (or a graph) is a
permutation σ of its vertices such that
every (strong) module of T is an interval of σ
Factoring permutation via a partition refinement algorithm in
linear time [de Mongolfier’03]
4 3 5 6 1 2 8 7

68. Modular decomposition algorithm for tournaments
A factoring permutation of a tournament T (or a graph) is a
permutation σ of its vertices such that
every (strong) module of T is an interval of σ
Factoring permutation via a partition refinement algorithm in
linear time [de Mongolfier’03]
4 3 5 6 1 2 8 7

69. Modular decomposition algorithm for tournaments
A factoring permutation of a tournament T (or a graph) is a
permutation σ of its vertices such that
every (strong) module of T is an interval of σ
Factoring permutation via a partition refinement algorithm in
linear time [de Mongolfier’03]
4 3 5 6 1 2 8 7
Modular decomposition tree from a factoring permutation in
linear time [Capelle’97]

70. Modular decomposition of undirected graphs
Ehrenfeucht et al’s modular decomposition algorithm
Common Intervals of permutations
Modular decomposition of tournaments
Kernelization algorithm for FAST

71. FAST: Feedback Arc Set in Tournament
A tournament T and an integer k
Find a set of at most k arcs whose reversal transform T into a
transitive tournament

72. FAST: Feedback Arc Set in Tournament
A tournament T and an integer k
Find a permutation σ of the vertices with at most k backward
edges
1 2 3 4 5 6 7 8

73. FAST: Feedback Arc Set in Tournament
A tournament T and an integer k
Find a permutation σ of the vertices with at most k backward
edges
1 2 3 4 5 6 7 8
NP-Complete [Alon’06] [Charbit et al.’07]
FTP [Raman, Saurabh’06] [Alon et al.’09]
(1 + )-approximation scheme [Kenyon-Mathieu, Schudy’07]

74. FAST (2)
Obs.: A tournament is transitive iff there is no (directed) triangle
1 2
8 3
7 4
6 5

75. FAST (2)
Obs.: A tournament is transitive iff there is no (directed) triangle
1 2
8 3
7 4
6 5

76. FAST (2)
Obs.: A tournament is transitive iff there is no (directed) triangle
1 2
8 3
7 4
6 5

77. FAST (2)
Obs.: A tournament is transitive iff there is no (directed) triangle
Rule 1 [irrelevant vertex] If a vertex v is not contained in any
triangle, then delete v
Rule 2 [sunflower] If there is an arc
belonging to more that k distinct triangles,
then reverse it and decrease k by 1

78. FAST (2)
Obs.: A tournament is transitive iff there is no (directed) triangle
Rule 1 [irrelevant vertex] If a vertex v is not contained in any
triangle, then delete v
A reduced tournament contains no source nor sink
Rule 2 [sunflower] If there is an arc
belonging to more that k distinct triangles,
then reverse it and decrease k by 1
The span s(− ) of a backward arc of a reduced tournament is
→
uv 2k + 2

79. FAST (3)
Rule 3 [acyclic module] Let M be a maximal acyclic module.
If there are at most p = |M| arcs from N + (M) to N − (M),
then reverse all these arcs and decrease k by p.

80. FAST (3)
Rule 3 [acyclic module] Let M be a maximal acyclic module.
If there are at most p = |M| arcs from N + (M) to N − (M),
then reverse all these arcs and decrease k by p.

81. FAST (3)
Rule 3 [acyclic module] Let M be a maximal acyclic module.
If there are at most p = |M| arcs from N + (M) to N − (M),
then reverse all these arcs and decrease k by p.
acyclic module Rule ⇒
ti2 s(− )
→
uv
ti

82. FAST (3)
Rule 3 [acyclic module] Let M be a maximal acyclic module.
If there are at most p = |M| arcs from N + (M) to N − (M),
then reverse all these arcs and decrease k by p.
sunflower Rule ⇒
ti2 s(− )
→
uv k(2k + 2)

83. FAST (3)
Rule 3 [acyclic module] Let M be a maximal acyclic module.
If there are at most p = |M| arcs from N + (M) to N − (M),
then reverse all these arcs and decrease k by p.
sunflower Rule ⇒
ti2 s(− )
→
uv k(2k + 2)
Theorem [Bessy et al.’09]: Every instance (T , k) of k-FAST can
be reduced in polynomial time to an equivalent instance (T , k )
such that √
|T | 2k + ti = O(k k) and k k

84. Kernelization algorithm
poly time
(G,k) (G’,k’)
|G’¦=f(k)
Given a parameterized instance (I, k) of a problem, a kernelization
algorithm computes in polytime an equivalent instance (I , k ) st.
k = f (k) and |I | g (k)

85. Kernelization algorithm
poly time
(G,k) (G’,k’)
|G’¦=f(k)
Given a parameterized instance (I, k) of a problem, a kernelization
algorithm computes in polytime an equivalent instance (I , k ) st.
k = f (k) and |I | g (k)
√
we described a O(k k)-vertex kernel for FAST based on
modular decomposition
best known result: O(k)-vertex kernel ([Bessy et al.’09], [P.,
Perez, Thomass´’11])
e

86. Kernelization algorithm
poly time
(G,k) (G’,k’)
|G’¦=f(k)
Given a parameterized instance (I, k) of a problem, a kernelization
algorithm computes in polytime an equivalent instance (I , k ) st.
k = f (k) and |I | g (k)
Other modular decomposition kernelizations:
O(k 2 )-vertex kernel for cluster editing
O(k 3 )-vertex kernel for cograph editing
also for min flip consensus tree, closest 3-leaf
power. . .

87. Some conclusions
Modular decomposition plays an important role in the context
of many graph classes
permutation graphs, interval graphs, comparability graphs . . .
even perfect graph)

88. Some conclusions
Modular decomposition plays an important role in the context
of many graph classes
permutation graphs, interval graphs, comparability graphs . . .
even perfect graph)
Many examples of partitive (weakly partitive) families are
known in various contexts
modules in undirected graphs, digraphs, hypergraphs
common interval of permutations

89. Some conclusions
Modular decomposition plays an important role in the context
of many graph classes
permutation graphs, interval graphs, comparability graphs . . .
even perfect graph)
Many examples of partitive (weakly partitive) families are
known in various contexts
modules in undirected graphs, digraphs, hypergraphs
common interval of permutations
Various generalizations
bimodular decomposition (module adapted to bipartite graphs)
bipartitive families : eg. split decomposition of graphs -
O(n + α(n, m).m) circle graph recognition
crossing families, union-difference families of sets. . .
clique-width (cographs are clique-widht 2 graphs), rankwidth

90. To learn / read more
Habib and P. ”A survey of the algorithmic aspects of modular
decomposition” in Computer Science Review 4:41-59, 2010

91. To learn / read more
Habib and P. ”A survey of the algorithmic aspects of modular
decomposition” in Computer Science Review 4:41-59, 2010
F. de Montgolfier, ”D´composition modulaire de graphes,
e
th´orie, extensions et algorithmes”, Phd Thesis (in French),
e
2003
B.M. Bui Xuan, ”Tree-representation of set families in graph
decompositions and efficient algorithms”, Phd Thesis, 2008

92. To learn / read more
Habib and P. ”A survey of the algorithmic aspects of modular
decomposition” in Computer Science Review 4:41-59, 2010
F. de Montgolfier, ”D´composition modulaire de graphes,
e
th´orie, extensions et algorithmes”, Phd Thesis (in French),
e
2003
B.M. Bui Xuan, ”Tree-representation of set families in graph
decompositions and efficient algorithms”, Phd Thesis, 2008
C. Crespelle, ”Repr´sentations dynamiques de graphes”, PhD
e
Thesis (in French), 2007 attend I. Todinca’s talk !!!

93. To learn / read more
Habib and P. ”A survey of the algorithmic aspects of modular
decomposition” in Computer Science Review 4:41-59, 2010
F. de Montgolfier, ”D´composition modulaire de graphes,
e
th´orie, extensions et algorithmes”, Phd Thesis (in French),
e
2003
B.M. Bui Xuan, ”Tree-representation of set families in graph
decompositions and efficient algorithms”, Phd Thesis, 2008
C. Crespelle, ”Repr´sentations dynamiques de graphes”, PhD
e
Thesis (in French), 2007 attend I. Todinca’s talk !!!
Common intervals and sorting by reversal:
attend M. Bouvel’s talk!!!